This is interesting history!
The earliest radios used no tubes at all, they were just fancy crystal radios. The first ones used a "galena whisker" as a diode.
When the tubes came they first needed batteries, because no one had AC yet, and in some places it was even DC. But when AC came to the home, there were companies that wanted everyone to buy their cheap radios, and they didn't want to use transformers because they were half the cost of the whole device. So they went through a period where they tried to power everything straight off the wall. And this meant, they had to wire the heaters for the tubes in series, because no 6.3 volts because no transformer.
So this is when cathodes became popular, because suddenly you had heaters sitting 50 volts above ground.
This idea of powering straight off the wall resurfaced several times. When the first 7 and 9 pin miniature tubes started coming out, there were a lot of compact radios from that period with no transformers, and tubes like 50C5, 35W4, and so on. Then the idea resurfaced again when the ultra-mini tubes came out and the radios got even smaller, because Eveready and Duracell weren't in all the 7-11's yet.
I remember building a geiger counter when I was 9 or so, it used a 45 volt battery to power the photo multiplier tube. I had to go the electronics store to buy one and they had to special order it.
Hm. Once I tried powering a 100 watt amp off my car, with an inverter. It lasted about half an hour.